Field Transformations: Connections to Symmetries

[maxverywell]

What are the differences in (scalar) field transformations:

1) \phi(x)\to \phi'(x)

2) \phi(x)\to \phi'(x')

3) \phi(x)\to \phi(x')

How this transformations are connected to internal and external symmetries?

For example, if we take spacetime global translations x^{\mu}\to x'^{\mu}=x^{\mu}+\epsilon^{\mu} which one of the 3 is the corresponding transformation of the field?

 

[nrqed]

What are the differences in (scalar) field transformations:

1) \phi(x)\to \phi'(x)

2) \phi(x)\to \phi'(x')

3) \phi(x)\to \phi(x')

How this transformations are connected to internal and external symmetries?

For example, if we take spacetime global translations x^{\mu}\to x'^{\mu}=x^{\mu}+\epsilon^{\mu} which one of the 3 is the corresponding transformation of the field?

A scalar field is invariant under Lorentz transformations. What this means is that

\phi(x) = \phi'(x')~~~~(1)

What this implies is that the field must transform,\phi(x)\to \phi'(x) in such a way that the field transformation compensates for the transformation of the coordinate.
To find the explicit form of \phi'(x) all you must do is to plug x' into Eq. (1) and Taylor expand.

 

[maxverywell]

A scalar field is invariant under Lorentz transformations. What this means is that

\phi(x) = \phi'(x')~~~~(1)

What this implies is that the field must transform,\phi(x)\to \phi'(x) in such a way that the field transformation compensates for the transformation of the coordinate.
To find the explicit form of \phi'(x) all you must do is to plug x' into Eq. (1) and Taylor expand.

Edit: thnx, I get it.

Is this valid only for real scalar fields?

I'm trying to prove energy-momentum conservation for space-time translations but this isn't proof for general case of energy-momentum conservation, only for scalar fields, but they don't exists in nature so how this can be useful?

 

[The_Duck]

Energy-momentum conservation comes from the invariance of the Lagrangian under translations. To express a field phi in a translated coordinate system, you used instead of the field phi(x) the field phi(x+a). This does not depend on whether phi is a scalar field or some higher spin field. On making this replacement this you find that your Lagrangian is unchanged, which leads to energy-momentum conservation.

A scalar field is defined by its behavior under Lorentz transformations: to express phi in a Lorentz-transformed (boosted or rotated) frame you replace
\phi(x) \to \phi(\Lambda^{-1} x)
Contrast a higher-spin field, which will have several components that mix under Lorentz transformations:
\psi_a(x) \to {D(\Lambda)_a}^b\psi_b(\Lambda^{-1}x)
However you will only need to start thinking about these more complicated transformations when you ask about the conservation laws that come from Lorentz symmetry, namely angular momentum conservation.